In terms of degree of anger, 憤怒
is higher 生气
.
In terms of context, 生气
is usually used in colloquial context, whereas 憤怒
is usually used in formal context.
Another term you may be interested in is 抓狂
, which is a colloquial term with degree of anger higher than 憤怒
Here is an interesting joke that can hopefully help you understand better.
I have quoted below the key phrases that denote the difference (forgive the offensive language used):
生气
「X的,神經病。你去死好了!」
"F***ing psycho, go to hell!"
憤怒
「你欠扁是不是?妈的,要是再打来,给我试试看!」
"You want some beef? F***, I dare you call again!"
抓狂
「他妈的!去你老母!你娘卡好!」
"F*** your mom!" (three variations with the same meaning)